Use of a single heated surface for the estimation of thermal conductivity components of orthotropic 3D solids

Figures & data

Figure 1 Geometry and coordinates.

Table I Sensor locations

Sensor	Sensor location
1	(0.5, 1, 0.5)
2	(1, 0.5, 0.5)
3	(0.25, 0.25, 0)
4	(0.5, 0.5, 0)
5	(0.75, 0.75, 0)

Figure 2 Normalized sensitivity coefficients for: (a) sensor 1; (b) sensor 2; (c) sensor 3; (d) sensor 4; (e) sensor 5.

Figure 3 Determinant of the information matrix for the: (a) sensor configuration 1; (b) sensor configuration 2; (c) sensor configuration 3; (d) sensor configuration 4.

Table II Configurations for the sensors

Configuration	Sensors
1	1 and 2
2	1, 2 and 3
3	1, 2 and 4
4	1, 2 and 5

Figure 4 Effects of the size of the heating surface on the determinant of the information matrix for t_f = t_h = 1 and sensor configuration 2.

Table III Test-cases and estimated parameters

Test-case	Arrangement	kx		ky		Kz
		Estimated value	99% Confidence interval width	Estimated value	99% Confidence interval width	Estimated value	99% Confidence interval width
1	1	1.00	0.02	14.96	0.55	15.01	0.05
2	1	1.00	0.02	15.04	0.55	15.00	0.05
3	1	1.00	0.02	15.07	0.55	15.00	0.05
4	2	16.35	7.40	14.35	6.27	1.00	0.01

Table

a, b, c	body dimensions along the x, y and z directions, respectively
FI	information matrix
I	number of measurements per sensor
Fx, Fy, Fz	normalized sensitivity coefficients for the thermal conductivity components along the x, y and z directions, respectively
J	sensitivity matrix
kx, ky, kz	thermal conductivity components along the x, y and z directions, respectively
M	number of sensors
P	vector of unknown parameters
S(P)	ordinary least-squares norm
th	heating time
tf	duration of the experiment
T	vector of estimated temperatures
Y	vector of measured temperatures

Table

σ	standard-deviation of the measurements
σj	standard-deviation estimated for parameter Pj

Table

*	dimensional variables
k	number of iterations

Table

i	refers to time ti
m	refers to sensor M